Let's solve the problem step by step.

Part (a) - List all possible rational zeros of $P(x)$ given by the Rational Zeros Theorem.

The Rational Zeros Theorem states that any rational zero of a polynomial $P(x)$ is of the form $\frac{p}{q}$, where:

• $p$ is a factor of the constant term (in this case, the constant term is $-2$).
• $q$ is a factor of the leading coefficient (in this case, the leading coefficient is $3$).

Step 1: Factors of the constant term ($-2$) The factors of $-2$ are $\pm 1, \pm 2$.

Step 2: Factors of the leading coefficient ($3$) The factors of $3$ are $\pm 1, \pm 3$.

Step 3: List all possible rational zeros Now, we combine all the factors of $p$ and $q$ to form possible rational zeros $\frac{p}{q}$.

The possible rational zeros are:

• $\pm 1, \pm \frac{1}{3}, \pm 2, \pm \frac{2}{3}$.

So, the possible rational zeros of the polynomial are: $\boxed{1, -1, \frac{1}{3}, -\frac{1}{3}, 2, -2, \frac{2}{3}, -\frac{2}{3}}.$

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Part (b) - From the graph, determine which of the possible rational zeros are actually zeros.

Let's analyze the graph based on the information provided:

• The curve changes direction at $(-1, 0)$, which means $x = -1$ is a zero.
• The curve crosses the x-axis again at $x \approx 2$, which means $x = 2$ is another zero.

From the given graph, the zeros appear to be $x = -1$ and $x = 2$.

So, the rational zeros that actually turn out to be zeros are: $\boxed{-1, 2}.$

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Let me know if you'd like further clarification or if you'd like to dive into the next steps! Here are some related questions:

1. How do you use synthetic division to test a potential zero?
2. How can you find the multiplicity of a root from the graph?
3. What is the significance of the graph changing direction at a zero?
4. How can you factor a polynomial completely once you know some zeros?
5. What does the end behavior of a polynomial function tell you about its degree?

Tip: When you're working with rational zeros, always test the potential zeros systematically by plugging them into the polynomial or using synthetic division!